Geodesic Congruences and Their Deformations in Bertrand Space-times

TL;DR

This paper investigates the properties of Bertrand space-times, focusing on energy conditions and geodesic deformations, and verifies geodesic focusing through numerical analysis of ESR parameters.

Contribution

It analyzes energy conditions and geodesic behaviors in Bertrand space-times, providing numerical verification of geodesic focusing and insights into their physical relevance.

Findings

Weak and strong energy conditions hold in certain regions.

Explicit demonstration of geodesic focusing.

Verification of Raychaudhuri equation through numerical analysis.

Abstract

We study the energy conditions and geodesic deformations in Bertrand space-times. We show that these can be thought of as interesting physical space-times in certain regions of the underlying parameter space, where the weak and strong energy conditions hold. We further compute the ESR parameters and analyze them numerically. The focusing of radial time-like and radial null geodesics is shown explicitly, which verifies the Raychaudhuri equation.